Any continuous function can be uniformly approximated by smooth functions.

I would like to have something similar - in what-ever sense - for continuous manifolds.

For example, by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\mathbb R^{2n}$. You can construct a continuous function $f$ with image $[-1,1]$ on $\mathbb R^{2n}$, whose zero level set is exactly (the image of) $M$.

A meaning of "approximating a manifold" would be to approximate such a level set function by smooth functions. However, Whitney's theorem is non constructive, you need a metric on the manifold for the question to make sense, and there are likely to appear difficulties.

Do you where to find a elaboration on questions like the above? (Of course, different approaches are of interest as well.). Thank you.

Gromov-Hausdorff metric would be an obvious one if you allow metrics. See the wikipedia page for details. IMO this is perhaps one of the most sensible approaches -- even in your function space example, one way or another a metric lurks in the background. If you remove metrics, do you consider diffeomorphic manifolds to be identical for these purposes, or do you want to allow something else?
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Ryan BudneyDec 13 '10 at 9:45

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There's also the "space of manifolds", things like the quotient $Emb(M, \mathbb R^\infty)/Diff(M)$, this is the space of submanifolds of $\mathbb R^\infty$ which are diffeomorphic to a given manifold $M$, with the Whitney/weak topology. So "approximation of a manifold" here would be described by a neighbourhood base of a given point in this embedding space.
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Ryan BudneyDec 13 '10 at 9:53

4 Answers
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A simply-looking meaning is the one of Gromov-Hausdorff, as mentioned by Ryan Budney.
It has many cool applications, but none that I'm aware of are to topological manifolds.

The kind of approximation that one normally uses to prove something about topological manifolds is representing your manifold as an inverse limit of polyhedra (rather than smooth manifolds). Examples of such use are incidentally given below.

I don't think that

by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\Bbb R^{2n}$.

Whitney's embedding theorem applies only to smooth manifolds. To embed a non-triangulable topological manifold in $\Bbb R^{2n+1}$ (note the dimension shift) one "approximates" the manifold by polyhedra and embeds those first (using PL general position); no simpler way is known. This is called the Menger-Nöbeling-Pontryagin embedding theorem (or after various subsets of the 3 authors); a clear proof appears in J. R. Isbell, Embeddings of inverse limits, Ann. of Math. 70 (1959), 73-84 and in Isbell's book "Uniform spaces". (Many other books give a less explicit proof based on the Baire category theorem.)

If you really want to embed a non-triangulable topological $n$-manifold $M$ in $\Bbb R^{2n}$ (and not just in $\Bbb R^{2n+1}$), this is harder. Earliest results that seem to imply this are in Bryant-Mio and Johnston's 1999 papers in Topology, available from Ranicki's website. A more direct (and I think much easier) proof is in this paper (based again on approximation by polyhedra).
In wondering about other possible approaches, I don't see how Kirby-Siebenmann could help: they show that $M\times\Bbb R$ is still not a PL manifold, if $n>4$; I don't know if it could be triangulable.

I doubt that this is what you are looking for, but I cannot resist mentioning this kind of beautiful approximation. One way to define a (riemannian) manifold is via a commutative version of Connes's spectral triples.

As shown by Roggenkamp and Wendland in this beautiful paper, there exist abstract conformal field theories which in a certain limit (typically a large volume limit) give rise to such commutative spectral triples. A expository version of their results can be found in this more recent preprint.

Now one can have maps from $T$ to each $M_i$, each map a homeomorphism
when restricted to an ever larger open set, in such a way that these
open sets exhaust $T$.

In general I suppose you'd want a net of manifolds of a fixed dimension and a corresponding
directed set of open sets that exhaust the manifold you mean to approximate. You wouldn't even have to have the limit manifold in advance if you had a suitable family of compatible maps connecting the various manifolds in your net.

I doubt you can, in any reasonable way, approximate the topology of a compact topological manifold by distinct compact topological manifolds of the same dimension.

You may also be interested in the geometric measure theory perspective on what is a nonsmooth surface supporting a "kind of differential geometry" (formulation from Frank Morgan's book Geometric measure theory: a beginner's guide). The relevant notion is that of "rectifiable currents."